(a) Find the sum of all multiples of 9 lying between 300 and 700.
OR
(b) The 26th, 11th and the last term of an A.P. are 0, 3 and -51 respectively. Find the common difference and the number of terms of the A.P.
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Step-by-Step Solution
Step 1: Formulate equations from given terms
We are given the 26th term (a26) and the 11th term (a11) of an Arithmetic Progression (A.P.). The general formula for the nth term of an A.P. is an=a+(n−1)d, where a is the first term and d is the common difference. Using this formula, we can set up two linear equations.
Step 2: Solve for common difference (d)
To find the common difference d, we can subtract the second equation from the first. This eliminates the variable a, allowing us to solve for d.
Step 3: Solve for the first term (a)
Now that we have the value of d, we can substitute it back into either of the original equations to find the first term a. We'll use the equation a+10d=3.
Step 4: Find the number of terms (n)
We are given that the last term of the A.P. is −51. Using the nth term formula again, an=a+(n−1)d, we can substitute the values of a, d, and an to solve for n, the total number of terms.